arXiv math/01041452001-04-12math.COmath.NT

New asymptotic bounds for self-dual codes and lattices

Eric M. Rains

备注：: 27 pages LaTeX, AMS macros

英文摘要：

We give an independent proof of the Krasikov-Litsyn bound d/n<~(1-5^{-1/4})/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(n^{1/2}).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Low dimensional strongly perfect lattices. III: Dual strongly perfect lattices of dimension 14

The extremal 3-modular lattice $[\pm G_2(3)]_{14}$ with automorphism group $C_2 \times G_2(\F_3) $ is the unique dual strongly perfect lattice of dimension 14.

2008-09-03math.COmath.NT

Low dimensional strongly perfect lattices IV: The dual strongly perfect lattices of dimension 16

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Λ_{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one pair of new lattices $Γ_{16}$ and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices.

2019-05-17math.COmath.NT

Some Extremal Self-Dual Codes and Unimodular Lattices in Dimension 40

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.

2012-03-22cs.ITmath.COmath.IT

Self-dual codes over $\mathbb{F}_5$ and $s$-extremal unimodular lattices

New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.

2022-08-07cs.ITmath.COmath.IT

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $U_q(sl_2)$

The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces of an abelian automorphism subgroup is shown to act as the adjacency matrix of a weighted subspace lattice. The connection between the latter and $U_q(sl_2)$ is used to find the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.

2021-08-31math.CO

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other classes of structures in everyday use, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriatelly chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokic have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the cl…

2015-11-24math.CO

更多相关论文收起更多论文

多面体的对偶格函数

Dual Lattice Functions of Polytopes

Peking University（北京大学）；University of Michigan（密歇根大学）；Colby College（科尔比学院）

研究通过有理多面体支撑函数指数的离散拉普拉斯变换定义对偶格函数，证明其可估值，乘以环面形式成指数多面体典范形式，推动环面多面体研究，是正几何离散化第一步。

2026-07-10math.CO

Optimal Self-Dual Z4-Codes and a Unimodular Lattice in Dimension 41

For lengths up to 47 except 37, we determine the largest minimum Euclidean weight among all Type I Z4-codes of that length. We also give the first example of an optimal odd unimodular lattice in dimension 41 explicitly, which is constructed from some Type I Z4-code of length 41.

2011-11-15cs.ITmath.COmath.IT

Duality and Outermost Boundaries in Generalized Percolation Lattices

In this paper we consider a connected planar graph $G$ and impose conditions that results in $G$ having a percolation lattice-like cellular structure. Assigning each cell of $G$ to be either occupied or vacant, we describe the outermost boundaries of star and plus connected components in $G$. We then consider the dual graph of $G$ and impose conditions under which the dual is also a percolation lattice. Finally, using $G$ and its dual, we construct vacant cell cycles surrounding occupied components and study left right crossings and bond percolation in rectangles.

2021-08-15math.COmath.PR

Non-rigidity degrees of root lattices and their duals

Non-rigidity degree of a lattice $L$, nrd$L$, is dimension of the L-type domain to which $L$ belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of $D_n^*$, and the case of $E_7^*$ are decided. We describe explicitly the $L$-type domain ${\cal D}(D_n^*)$, $n \ge 4$. For $n$ odd, it is a non-simplicial polyhedral open cone of dimension $n$. For $n$ even, it is one-dimensional, i.e. for even $n$, $D_n^*$ is an edge form.

2002-02-11math.COmath.GT

Percolation on dual lattices with k-fold symmetry

Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particula…

2007-02-06math.COmath.PR

On combinatorics of Voronoi polytopes for perturbations of the dual root lattices

The Voronoi conjecture on parallelohedra claims that for every convex polytope $P$ that tiles Euclidean $d$-dimensional space with translations there exists a $d$-dimensional lattice such that $P$ and the Voronoi polytope of this lattice are affinely equivalent. The Voronoi conjecture is still open for the general case but it is known that some combinatorial restriction for the face structure of $P$ ensure that the Voronoi conjecture holds for $P$. In this paper we prove that if $P$ is the Voronoi polytope of one of the dual root lattices $\mathsf{D}_d^*$, $\mathsf{E}_6^*$, $\mathsf{E}_7^*$ or…

2021-12-16math.COmath.MG

On the Asymptotics of the Finite-Perimeter Partition Function of Two-Dimensional Lattice Vesicles

We derive the dominant asymptotic form and the order of the correction terms of the finite-perimeter partition function of self-avoiding polygons on the square lattice, which are weighted according to their area A as q^A, in the inflated regime, q>1. The approach q->1^+ of the asymptotic form is examined.

1998-09-10cond-mat.stat-mechmath-phmath.CO